In order to answer these questions, you have to understand the path that oil takes from the ground to your gas tank. "Oil companies sell to refiners. Refiners sell to distributors. Distributors sell to retailers." Whoa. That's a lot of steps. Let's break it down.

Oil companies sell to refineries

Not all oil companies own refining companies. In fact, a very few actually own their refineries. They just find, drill and produce crude which they sell on the NYMEX or other markets, to refiners. A refiner can take crude from anybody—Exxon from Shell, Chevron from BP, etc.

Refineries sell to distributors

This means that an Exxon refinery can sell gasoline to a Shell distributor. The distributor chooses where to buy his products from. Naturally, Shell Oil Refining would offer a Shell Distributor a pretty discounted price, but not necessarily the lowest he might find. Thus distributors can buy from anybody and sell to anybody, regardless of their "affiliation" with a given oil company. A distributor may be an "Authorized Chevron Distributor", and thus he can sell Techron type gasoline. But that doesn't mean he is restricted to just selling Chevron products. Independent distributors (more of them than any other type) are listed in the Yellow Pages under Fuel Companies or Fuel Distributors. These are the middlemen.

Distributors sell to retailers

These independents actually buy gas from whoever they get the best deal from, then sell it to their customers. If you have ever wondered why there is an Exxon tanker filling up a Fina gas station tank, it is because the distributor is an Exxon affiliate, but the Fina station owner got a better deal from Exxon than from Fina. This happens all the time. Distributors also sell grease and transmission fluid and hydraulic fluid and motor oil to industry. You can often see distributors trucks servicing construction sites or businesses operating a fleet of vehicles.
Retailers can buy whatever they want from whoever they can get the best price! These corporations are usually very divorced from the nuts and bolts of oil companies, that focus on retailing gasoline and convenience store operation. The retail chain may be a Shell-owned convenience store, supplied by a Shell affiliated distributor, but he (distributor) may buy his oil from Murphy Oil Refining. So while the buyer thinks he is getting Shell gasoline, what he is getting is gasoline from a Shell Distributor, but produced by Murphy Oil Refining.
I have loosely used Big Oil Company names here to make a point. However, many of the big guys keep a close reign on their distributors. But that does not preclude them selling whatever they want to independent retailers, like Samir's Quicky Mart.
Thus unless a gasoline has a patented ingredient (like Techron) or other additive, it will not necessarily wind up being sold by the affiliate retailer. And we haven't even gotten to supply contracts yet, which further complicate things.

Summing up...

At every level of these distribution chains, people are trying to buy low and sell high. This is extremely confusing for everybody, but it is the way it works. But if you keep this in mind—oil companies sell to refiners—you can quickly see why oil companies get pissed when the finger pointing starts.

So, if there is gouging, where is it happening?

My opinion is that the gouging, if any, is going on at a very local level, no higher up than the local gasoline distributors. The most opportunistic way to maximize your cash would be to buy this week's gasoline for $2.50 a gallon, and then if something (like high oil prices) allows you to, mark this shipment of $2.50 gas up to $3.00. [Keep in mind that some states regulate the maximum profit on a gallon of gas.] Normally gas markup is about a nickel a gallon or such. The convenience store makes much more on food—the gasoline is the draw to get them into the store to buy cigarettes or candy or a coke, which have much higher profit margins.
The people who control the final price at the pump are the retailing companies or independent store owners. And these guys are more than happy to put it off on the oil companies, as they are very removed from them!
I know—as clear as mud. But it is the "free economy" at work...

Why are oil companies making record profits?

Because the wells they are producing from today were drilled in past years, where they used $15-25 per barrel as their estimated selling price for the oil. Thus when the market got tight, they have cheap oil going at a higher price. Why? Simple—demand is high.
Now this higher priced oil is filtering back to them in the form of higher priced goods, so the profits decline slowly over time as higher priced energy enters the economic mill. But right now, they are making a lot of money.

So, it is not the case that oil companies are incurring high exploration costs and passing these costs onto the consumer?

No. Oil companies are not setting the retail price except via the NYMEX, where they sell to refiners.
When the traders run the price up, oil companies naturally win. But the crude price is set by market demand i.e., what will they pay today?

So, let's say, very hypothetically, that Exxon Mobil wanted to make a gesture of good faith. Could they sell the crude for cheaper in hopes of lower prices at the pump, or does the nature of the futures market and all of the middlemen make that impossible?

It would not work for exactly the reasons you think.
But...
ExxonMobil, owning their own up and down stream divisions, could sell at a loss or reduced profit on the retail end, provided they compensated their convenience store owners for their lost gasoline revenues (these stores are franchises). But that would make whatever cut they did offer twice as financially painful—they would take the announced cut and associated reduction in profit, and then have to pay the store owners their traditional profit to keep them happy.
So you are not asking them to just fall on their own sword, but to get back up and hurl their bloodied body on it again...ouch!
So—if ExxonMobil did do this, it would be a huge gesture! But only those in the same business would understand the magnitude of what they had done. And whoever did it would shortly be replaced by the Board of Directors as the principal shareholders all called for his head on a pike!